Sensor with movable parts and biasing

ABSTRACT

Methods and apparatuses are provided wherein a sensor which comprises at least two electrodes and a movable part is alternately biased with at least two different voltages.

RELATED APPLICATION

This application is a divisional of U.S. patent application Ser. No.13/035,896, filed Feb. 25, 2011 (now U.S. Pat. No. 9,525,925), which isincorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates to sensors comprising a movable part andto readout and amplification of a signal output by the sensor.

BACKGROUND

Sensors with movable parts are used in many applications, for example asacceleration sensors or as sound sensors, i.e. microphones. In sometypes of these sensors, a movable object is displaced with respect totwo or more electrodes arranged close to the moving object, and changeof capacitance between the movable object and the electrodes caused bythis movement may be read out from the sensor. For example, the movableobject may be a membrane of a microphone.

Such sensors may for example be implemented in the form ofmicroelectromechanical systems (MEMS), which are also referred to asmicro machines or systems based on microsystems technology.

As the signal provided by the sensor in many cases is comparativelyweak, the signal is usually amplified before it is further processed.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows a block diagram of an apparatus according to an embodiment.

FIG. 2 shows an example for a sensor used in some embodiments.

FIG. 3 shows a circuit diagram of an apparatus according to anembodiment.

FIG. 4 shows a flow chart illustrating a method according to anembodiment.

FIG. 5 shows a curve for illustrating some features of some embodiments.

FIG. 6 shows a block diagram of an apparatus according to an embodiment.

FIGS. 7A and 7B, collectively referred to hereinafter as FIG. 7, show acircuit diagram of an apparatus to an embodiment.

DETAILED DESCRIPTION OF THE INVENTION

In the following, some embodiments of the present invention will bedescribed in detail. It is to be understood that the followingdescription is given only for the purpose of illustration and is not tobe taken in a limiting sense. The scope of the invention is not intendedto be limited by the embodiments described hereinafter with reference tothe accompanying drawings, but is intended to be limited only by theappended claims and equivalents thereof.

It is also to be understood that in the following description ofembodiments any direct connection or coupling between functional blocks,devices, components, circuit elements or other physical or functionalunits shown in the drawings or described herein could also beimplemented by an indirect connection or coupling, i.e. a connection orcoupling comprising on or more intervening elements. Furthermore, itshould be appreciated that functional blocks or units shown in thedrawings may be implemented as separate circuits in some embodiments,but may also be fully or partially implemented in a common circuit inother embodiments. In other words, the use of different functionalblocks or units in the drawings is intended to give the clearunderstanding of various functions performed by the correspondingapparatus, but is not to be construed as indicating that the functionalblocks have to be implemented as separate physical units.

It is further to be understood that any connection which is described asbeing wire-based in the following specification may also be implementedas a wireless connection unless noted to the contrary.

It should be noted that the drawings are provided to give anillustration of some aspects of embodiments of the present invention andtherefore are to be regarded as schematic only. In particular, theelements shown in the drawings are not necessarily to scale with eachother, and the placement of various elements in the drawings is chosento provide a clear understanding of the respective embodiment and is notto be construed as necessarily being a representation of the actualrelative location of the various components in implementations of therespective embodiment.

The features of the various embodiments described herein may be combinedwith each other unless specifically noted otherwise. On the other hand,describing an embodiment with a plurality of features is not to beconstrued as indicating that all those features are necessary forpracticing the present invention, as other embodiments may comprise lessfeatures and/or alternative features.

In the following embodiments, sensors having movable parts and at leasttwo electrodes are described. With such a sensor, the movable part isdisplaced with respect to the electrodes due to an event to be monitoredby the sensor, and this displacement causes a change of capacitancebetween the electrodes and the movable object indicative of the event,which can then be detected electrically. For example, in case of amicrophone, i.e. a sound sensor, the event may be an incoming soundwave, and in case of an acceleration sensor the event may be the sensorbeing accelerated. The electrodes may be biased with respect to themovable part, which may be effected by applying a voltage to theelectrodes, the movable part or both. An example for such a sensor willbe explained later in greater detail.

Turning now to the Figures, in FIG. 1 a block diagram according to anembodiment of the present invention is shown. The apparatus shown inFIG. 1 comprises a sensor 10, which in the embodiment of FIG. 1 is asensor comprising a movable part arranged adjacent to, for examplebetween, at least two electrodes. An output of sensor 10 is coupled withreadout circuitry 11, which outputs an output signal out. Sensor 10 mayfor example be a microphone or an acceleration sensor, but is notlimited thereto.

Furthermore, FIG. 1 comprises biasing circuitry 12 configured to biasthe electrodes of sensor 10 with respect to the movable part alternatelywith at least two different voltages Vm1, Vm2, for example by applyingthe voltages to the electrodes and/or to the movable part. Biasingcircuitry 12 may be coupled with readout circuitry 11. For example, inan embodiment readout circuitry 11 may change the readout or perform thereadout depending on the switching between the voltages Vm1 and Vm2. Inanother embodiment, readout circuitry 11 may additionally oralternatively control biasing circuitry 12 regarding the switchingbetween the voltages Vm1, Vm2 and/or the magnitude of these voltages,thus providing a feedback path from sensor 10 via readout circuitry 11and biasing circuitry 12 back to sensor 10.

Through alternately biasing the first electrode and the second electrodeof sensor 10 with voltage Vm1 and voltage Vm2 and selecting the voltagesappropriately as will be explained further below, an amplification ofthe sensed signal may be obtained. This amplification, which may be seenas a type of superregenerative amplification, exploits the fact that asensor as described above may be approximated as a second order linearsystem which has a free response with an envelope y(t) that can beapproximated by

$\begin{matrix}{{y(t)} = {{y(0)} \cdot {\mathbb{e}}^{{- \frac{t - t_{dl}}{\tau}}\;}}} & (1)\end{matrix}$

wherein τ is a time constant and t_(dl) is a delay time. The timeconstant τ can be either positive (stable state) or negative (unstablestate). A stable state of the movable mass is a state where at leastabsent external forces like acoustic forces the movable part returns toan equilibrium position, while in its unstable state a displacement fromthe equilibrium position increases. The time constant τ may be varied byapplying different voltages to the electrode of a sensor. In particular,depending on the choice of voltage, the time constant τ may be madepositive or negative. By periodically switching between a positive and anegative time constant at a so called quenching frequency, anamplification may be obtained. The quenching frequency for anarrangement like the one shown in FIG. 1 may be increased using afeedback like the one mentioned above.

In FIG. 2, an example for a sensor in form of a microphone is shown. Themicrophone of FIG. 2 may be manufactured as a microelectromechanicalsystem (MEMS), for example based on a silicon substrate.

The microphone shown in FIG. 2 comprises an upper back plate 20 and alower back plate 21 to which a voltage may be applied, i.e. they may beused as first and second electrodes, respectively.

A membrane 22 is located between upper back plate 20 and lower backplate 21. In a zero position the distance between membrane 22 and upperback plate 20 is ×0, 1, and the distance between membrane 22 and lowerback plate 21 is ×0, 2. Due to mechanical and/or electrical forces,membrane 22 may be displaced, for example to a location where adisplaced membrane 22A is shown. The displacement is designated x′₀.

If the microphone of FIG. 2 is used in the embodiment of FIG. 1 thevoltages Vm1, Vm2 may be alternately applied to upper back plate 20 andlower back plate 21, i.e. at one time the voltage Vm1 is applied toupper back plate 20 and lower back plate 21, and at a different time thevoltage Vm2 is applied to upper back plate 20 and lower back plate 21,or may be alternately applied to membrane 22.

An electrical force between the displaced membrane 22A and the upperback plate 20 is labeled Fe,1, and an electric force between displacedmembrane 22 a and lower back plate 21 is labeled Fe,2. Fa is a forcecaused by an acoustic sound wave, and Fm is a mechanical force which maybe symbolized by a spring 23 with spring constant k_(mcch). It is to benoted that in such systems usually no real “spring” is present, but forexample the membrane may be suspended between mountings, and thus adisplacement of the membrane from its zero position causes a restoringforce.

V₁ is a voltage between upper back plate 20 and (displaced) membrane22A, and V₂ is a voltage between lower back plate 21 and (displaced)membrane 22A. The arrows next to voltages, distances and forces indicatethe “positive” direction of the respective quantity, i.e. a quantity inthe direction of the arrow is positive in the following equations andexplanations, and a quantity in the opposite direction is negative.

For simpler equations, a common mode voltage V₀ and a differential modevoltage V_(d) is defined such thatV ₁ =V _(c) +V _(d)/2  (2)V ₂ =V _(c) −V _(d)/2  (3)

The duration of a rising phase, i.e. a phase where the system isunstable, will be referred to as t_(r), and the duration of a decayingphase, i.e. a phase where the system is stable, will to referred to ast_(d), and the above-mentioned quenching frequency will be labeledf_(q)=1/T _(q), wherein T_(q)>t_(r)+t_(d).

Next, it will be described in some more detail how the above-explainedprinciples may be implemented for a microphone like the one shown inFIG. 2 or a similar sensor where a movable object like a membrane movesbetween two electrodes. The dynamics of such a system can beapproximated by a second order system. The system dynamics depend on thebiasing condition of capacitors formed by the electrodes and the movablepart. For large biasing resistances, the charge stored inside thecapacitor can be considered constant, and the dynamics are approximatelyindependent of the applied voltage. If, however, the biasing resistancesare small, the voltage across the capacitor plates (for example theelectrodes like upper back plate 20 and lower back plate 21 as well asthe membrane 22) can be considered constant, and the system dynamicschange with the bias voltage. The system can at least approximately bedescribed by the transfer functions H_(d)(s) and H_(e) (s) according to

$\begin{matrix}{{{H_{a}(s)} = \frac{- \frac{F_{n}}{m}}{s^{2} + {s\;\frac{r}{m}} + \frac{k^{\prime}}{m}}}{and}} & (4) \\{{{H_{e}(s)} = \frac{- \frac{F_{c}}{m}}{s^{2} + {s\;\frac{r}{m}} + \frac{k^{\prime}}{m}}},} & (5)\end{matrix}$

H_(a) (s) describing the membranes displacement as a function ofacoustic forces and H_(e) (s) describing the membranes displacement as afunction of electrostatic forces. For ease of reference, the microphoneshown in FIG. 2 will be used for the following explanations. It is,however, to be understood that the principles explained below may alsobe applied to other sensors where a movable object moves between atleast two electrodes, for example acceleration sensors.

m is the membrane's effective mass, r is a viscous damping, andk′=k_(mech)−k_(e1) (V_(c)) represents the effective spring constant,k_(mech) being the mechanical spring constant and k_(e1) being the“spring constant” caused by electrostatic forces and being dependent onthe bias voltage.

The physically possible free responses of such a device follow anenvelope summarized in the table below:

Type τ t_(dl) condition rising$\frac{2\; Q}{\omega_{0}\left( {1 - \xi} \right)}$$\tau\;{\ln\left( {\frac{1}{2} + \frac{1}{2\;\xi}} \right)}$ ξ > 1decaying $\frac{2\; Q^{\dagger}}{\omega_{0}}$ 0^(†) $Q > \frac{1}{2}$decaying $\frac{2\; Q^{\ddagger}}{\omega_{0}\left( {1 - \xi} \right)}$$\tau\;{\ln\left( {\frac{1}{2} + \frac{1}{2\;\xi}} \right)}^{\ddagger}$ ξ < 1, ^(†)oscillatory result. ^(‡)not valid for ξ → 0.

wherein

$\begin{matrix}{{\omega_{0}^{2} = \frac{k^{\prime}}{m}};{\frac{\omega_{0}}{Q} = \frac{r}{m}};{{\xi{\sqrt{1 - {4Q^{2}}}}} = {\sqrt{1 - {4\frac{k^{\prime}m}{r^{2}}}}}}} & (6)\end{matrix}$

By changing V_(c), for example by applying different bias voltage toupper back plate 20 and lower back plate 21 and/or to membrane 22 inFIG. 2, k′ is changed, and therefore the free response envelope of thesystem is either exponentially rising or decaying. This can be used insome embodiments to essentially use the microphone itself as anamplifier.

In FIG. 3, an embodiment of a circuit using these principles are shown.The circuit of the embodiment of FIG. 3 is a circuit not using feedback.

In the circuit diagram of FIG. 3, a sensor 32 is represented by twovariable capacitances 33, 34. Capacitance 33 may be the capacitancebetween a first electrode and a moving object, for example between upperback plate 20 and membrane 22, and capacitance 34 may be a capacitancebetween a second electrode and the moving object, for example betweenlower back plate 21 and membrane 22.

In the embodiment of FIG. 3, ϕ₁ and ϕ₂ are two phases of anon-overlapping clock which control two switches 30, 31, respectively.In other words, switches 30, 31 are alternately closed, thus applyingeither a voltage Vm1 or a voltage Vm2 to the movable object of sensor32, for example to a membrane. A common mode feedback circuitry 310which is fed an (input) common mode voltage Vcmin biases the electrodes,for example, upper back plate 20 and lower back plate 21 of thearrangement of FIG. 2. Coupled to the electrodes of sensor 32 is adifferential amplifier 37. Parallel to differential amplifier 37, onboth sides capacitors 36, 38 and switches 36, 39 are provided as shownin FIG. 3. Switches 35, 39 are switched depending on signal ϕ₂, i.e.these switches are closed when also switch 31 is closed. At outputs 311,312 an output voltage Vout may be tapped.

Next, the operation of the embodiment shown in FIG. 3 will be describedin some more detail. For the following explanation, a sensor like theone shown in FIG. 2 will be used, however, it is to be understood thatother sensors having a movable object between two electrodes may also beused.

The two voltages Vm1, Vm2 are chosen such that when voltage Vm1 isapplied the system becomes unstable and when voltage Vm2 is applied thesystem is stable. This can be obtained by setting k_(e1) (V_(c))according to

$\begin{matrix}{{k_{c\; 1}\left( V_{c} \right)} = \frac{4{zV}_{c}^{2}}{x_{0}^{3}}} & (7)\end{matrix}$

Such that it is greater than or smaller than k_(mcch), respectively,such that the effective spring constant k′ becomes negative or positive,respectively.

z in equation (7) is equal to

${z = \frac{ɛ_{0}R^{2}{\pi\left( {1 - \frac{r_{h}^{2}\rho_{h}}{\left( {r_{h} + x_{0}} \right)^{2}}} \right)}}{2}},$wherein ε₀ is the dielectric constant, R is the radius of the in thisexample circular back plate electrodes and/or the membrane in FIG. 2,and the term in brackets is a correctional factor which is approximatelyone, r_(h) being a radius of holes in the upper and lower back platesand p_(h) being a density of such holes. Such holes for example allowsound to reach the membrane. In other embodiments, no holes may be used,and the corresponding term may be omitted. In still other embodiments,non-circular back plate electrodes and/or membranes may be used, inwhich case in the above expression for z the term R²π is replaced by thearea of the back plate electrodes and/or membrane.

In other words, when signal Φ₂ closes switch 31, the voltages V₁ and V₂are set to Vm2 which is smaller than the so-called pull-in voltage, i.e.the voltage necessary to create an unstable system. Thus the system isstable, and the membrane displacement x′₀ is a result of the acousticforce Fa acting against the mechanical restoration force Fm.

When now signal Φ₁ closes switch 30 and on the other hand switch 31 isopened, the voltage Vm1 is used as a biasing voltage which is above thepull-in voltage such that the system becomes unstable and the membranestarts to increase its displacement exponentially. In other words, adisplacement caused by an acoustic force while switch 31 was closed isnow amplified when switch 30 is closed.

In the embodiment of FIG. 3, the change of the bias voltage from Vm2 toVm1 and back is a common mode signal that is not amplified by thedifferential amplifier 37 provided. On the other hand, the membranedisplacement changes the capacitances 33, 34 of the sensor in oppositedirections. The resulting charge difference is converted to a voltageacross the capacitances 36, 38 which are configured as feedbackcapacitors in FIG. 3. The output voltage Vout is then

$\begin{matrix}{{V_{out}\left( t_{r} \right)} = {\frac{2\; C_{0}V_{c}}{C_{F}x_{0}}{x_{0}^{\prime}\left( t_{r} \right)}}} & (8)\end{matrix}$

wherein C_(F) is the capacitance of capacitances 36, 38, C₀ is thenominal MEMS capacitance. It should be noted that in equation (8), thecapacitance change has been linearized, which is a good approximation.

The gain of the amplification provided by this concept is effectedduring the rising phase and depends on the characteristics of the sensorand the duration of the rising phase t_(r). The membrane displacement atthe end of the rising phase is

$\begin{matrix}{{{x_{0}^{\prime}\left( t_{r} \right)} = {{x_{0}^{\prime}(0)} \cdot \underset{\underset{gain}{︸}}{{\mathbb{e}}^{- \frac{t_{r} - t_{{dl},r}}{\tau_{r}}}}}},} & (9)\end{matrix}$

where x₀′ (0)=−F_(a)/k′ is the displacement at the beginning of therising phase, t_(dl,r) is a delay in the rising phase and τ_(r) is atime constant for the rising phase. Examples for the values of theseparameters and also the parameter ξ for the rising phase and thedecaying phase are shown in the table below.

Phase V_(c) [V] τ [s] t_(dl) [s] ξ rising 10 −8.5371e−6 925.4586e−91.2586 decaying 0 27.3292e−6  1.1753e−6 919.2163e−3

It should be noted that these numerical values serve only asillustration and may vary from sensor to sensor.

It should be noted that depending on the values of t_(r) and t_(d) theremay be a memory effect, i.e. a current initial displacement x₀′ (0) mayinfluence a following initial displacement in the next decaying phase.The influence may be estimated by a discrete time system with thefollowing impulse response

$\begin{matrix}{{h_{i}\lbrack n\rbrack} = {\left( {\mathbb{e}}^{{- \frac{t_{r} - t_{{dl},r}}{\tau_{r}}} - \frac{t_{d} - t_{{dl},d}}{\tau_{d}}} \right)^{n} = a^{n}}} & (10)\end{matrix}$

wherein a is a parameter. It should be noted that this represents anestimation only as for this equation initially non-moving membranes areassumed, however, at the transition from rising to decaying phase themembrane usually is moving.

The z-transform of the impulse response of equation (10) is

$\begin{matrix}{{H_{i}(z)} = \frac{1}{1 - {az}^{- 1}}} & (11)\end{matrix}$

which corresponds to the impulse response of a low pass filter/lossyintegrator. A parameter a equal to zero would lead to no memory, whichwould be the ideal case for a pure amplifier. However, in someembodiments, an integrating functionality, for example as part of adelta sigma modulator, may be desirable and implemented, for example ifthe stable bias voltage, i.e. the bias voltage in the decaying phase,has a differential component force feedback, which will be describedlater.

To achieve a stable operation, in an embodiment the parameter a is setto be smaller than one. This leads to the following constraint:

$\begin{matrix}{{{- \frac{t_{r} - t_{{dl},r}}{\tau_{r}}} < \frac{t_{d} - t_{{dl},d}}{\tau_{d}}},} & (12)\end{matrix}$

which puts a constraint on the values t_(r), t_(d) and therefore, asexplained above, limits the maximum achievable quenching frequency asT_(q) is greater than t_(r)+t_(d). The quenching frequency determinesthe frequency of readout, or, in other words, the sampling frequency. Onthe other hand, limiting t_(r) may limit the gain. Therefore, dependingon the application, the parameters may be selected to either achieve ahigh gain or a high quenching frequency, depending on the necessities.

For such a small parameter a, a robust operation may be obtained in someembodiments as the membrane will return to an equilibrium position andeven recover from a potential dynamic pull-in.

To give an example for the thus caused relationship between gain andquenching frequency, FIG. 5 shows the estimated quenching frequency forthe embodiment of FIG. 3 versus gain.

In the embodiment of FIG. 3, the system dynamics are changed byswitching the bias voltage between two discrete levels, being simple toimplement. In other embodiments, a switching between more than twovoltages may be implemented.

In FIG. 4, an embodiment of a method is schematically shown. The methodshown in FIG. 4 may be implemented in the embodiment of FIG. 1 or 3and/or may be implemented using the sensor shown in FIG. 2, but may alsobe implemented independently therefrom, for example with other sensorslike acceleration sensors or with other circuitry connected to thesensor.

At 40, a sensor with a movable part between at least two electrodes, forexample in form of a microelectromechanical system, is provided.

At 41, alternately a first voltage and a second voltage is appliedbetween the electrodes and the movable part.

At 42, the micro electromechanical system is read out, for example whilethe first voltage is applied and/or while the second voltage is applied.

As already mentioned, with respect to the embodiment of FIG. 1, in someembodiment a feedback may be provided. While in the embodiments of FIGS.3 and 4 no such feedback is provided, in other embodiments such afeedback may be present. For example, at 41 in FIG. 4 the alternatelyapplying of the first voltage and second voltage may depend on afeedback loop. Further embodiments using feedback will be described nextwith reference to FIGS. 6 and 7.

In FIG. 6, a block diagram of an apparatus according to an embodimentcomprising a feedback loop is shown. A sensor 60, in the embodiment ofFIG. 6, a microelectromechanical system comprising a movable part and atleast two electrodes, is provided. The sensor 60 receives an externalforce, in case of a microphone an acoustic force F_(a), and an electricforce due to biasing of the electrodes with respect to the movable part.These forces are added as symbolized by an adder 61 and cause mechanicaldynamics of the system, i.e. movement of the movable part, as symbolizedby block 2. The mechanical dynamics 62 result in a displacement x₀′,which, due to capacitance and therefore charge changes based on themovement, are converted to a voltage Vout as symbolized by a block 64.

The voltage Vout represents the sensed quantity and may be furtherprocessed, and is further fed to a controller 65 which generates abiasing voltage V_(d) which at least partially depends on Vout. Thevoltage V_(d) results in the electric force Fe as symbolized by a block63 converting a voltage to the force.

It should be noted that the various blocks shown in FIG. 6 symbolizevarious functions, but are not to be construed as being actual entitiesin the system. For example, the forces acting on the movable part like amembrane are added automatically due to the laws of mechanics, and noexplicit adder 61 is present, adder 61 just symbolizing this mechanicalfact. Likewise, forces acting on a movable part “automatically” lead tomechanical dynamics as symbolized by block 62, and no specific entity isneeded for this result. Similar considerations may hold true for otherblocks, for example blocks 63 and 64.

To explain the operation of an embodiment with feedback loop like theembodiment of FIG. 6, a general state space model for the sensor, inthis case microelectromechanical system 60, of the formq(t)=A _(q)(t)+B _(u)(t)  (15)y(t)=C _(q)(t)+D _(u)(t)  (16)

with

$\begin{matrix}{{q(t)} = {{\begin{bmatrix}x_{0}^{\prime} \\\overset{.}{x_{0}^{\prime}}\end{bmatrix}\mspace{14mu}{u(t)}} = \begin{bmatrix}{F_{a}(t)} \\{{V_{c}(t)}{V_{d}(t)}}\end{bmatrix}}} & (17)\end{matrix}$

will be used. The parameters A, B, C and D are defined as follows

$\begin{matrix}{A = {{\begin{bmatrix}0 & 1 \\{- \frac{k^{\prime}}{m}} & {- \frac{r}{m}}\end{bmatrix}\mspace{14mu} B} = \begin{bmatrix}0 & 0 \\{- \frac{1}{m}} & {- \frac{2z}{{mx}_{0}^{2}}}\end{bmatrix}}} & (18) \\{C = {{\begin{bmatrix}1 & 0\end{bmatrix}\mspace{14mu} D} = 0}} & (19)\end{matrix}$

The quantities not explained here have the same meaning as alreadyexplained with respect to FIGS. 1 to 5.

To illustrate the effects of the feedback loop, it is assumed thatcontroller 65 uses the state variables membrane displacement andvelocity of the membrane, i.e. x′₀ and {dot over (x)}′₀ or estimationsthereof, {dot over (x)}′₀ being the derivate of x′₀ with respect totime, for the determination of the feedback, i.e. the determination ofV_(d) or variation thereof in FIG. 6. If corresponding gains, i.e.“amplifications” of x_(c)′ and {dot over (x)}′₀ for generating thefeedback, are called k_(d) and k_(v), respectively, then a new systemmatrix A_(f) replacing matrix A above can be written as

$\begin{matrix}{A_{f} = \begin{bmatrix}0 & 1 \\{{- \frac{k^{\prime}}{m}} + k_{d}} & {{- \frac{r}{m}} + k_{v}}\end{bmatrix}} & (20)\end{matrix}$

By defining

$\begin{matrix}{{\omega_{0}^{2} = {\frac{k^{\prime}}{m} - k_{d}}};{\frac{\omega_{0}}{Q} = {\frac{r}{m} - k_{v}}}} & (21) \\{\xi = {{\sqrt{1 - {4Q^{2}}}} = {\sqrt{1 - {4\frac{m\left( {k^{\prime} - {mk}_{d}} \right)}{\left( {r - {k_{v}m}} \right)^{2}}}}}}} & (22)\end{matrix}$

(these definitions replace those of equations (6) for the currentanalysis of the feedback loop), closed loop transfer functions may bederived as

$\begin{matrix}{{H_{a}(s)} = {\frac{- \frac{F_{a}}{m}}{s^{2} + s^{\frac{\omega_{0}}{Q}} + \omega_{0}^{2}}\mspace{14mu}{and}}} & (23) \\{{H_{a}(s)} = \frac{- \frac{F_{a}}{m}}{s^{2} + s^{\frac{\omega_{0}}{Q}} + \omega_{0}^{2}}} & (24)\end{matrix}$

It can be seen from equation (21) that the values of ω₀ ² and ω₀/Q canbe chosen arbitrarily by adjusting the gains k_(d), k_(v), accordingly,for example by designing controller 65 of FIG. 6 appropriately. Thus,free response parameters and their time constants can be chosenarbitrarily as well, giving corresponding freedom in design of thesystem. Assuming for example zero membrane speed (x₀′=0), the freeresponse has the form

$\begin{matrix}{\frac{x_{0}^{\prime}(t)}{x_{0}^{\prime}(0)} = {{{\mathbb{e}}^{{- \frac{\omega_{0}}{2Q}}t}{\cosh\left( {\frac{\omega_{0}\sqrt{1 - {4Q^{2}}}}{2Q}t} \right)}} + {{\mathbb{e}}^{{- \frac{\omega_{0}}{2Q}}t}\frac{1}{\sqrt{1 - {4Q^{2}}}}{\sinh\left( {\frac{\omega_{0}\sqrt{1 - {4Q^{2}}}}{2Q}t} \right)}}}} & (25)\end{matrix}$

In some embodiments a sensor may have separate drive capacitances, i.e.capacitances which may be biased, and sense capacitances, i.e.capacitances for reading out the sensors. Such separate drive and sensecapacitances are for example of a use in accelerometers or gyroscopes.On the other hand, in microphones like the ones shown in FIG. 2 oftencapacitances which are used both for driving and sensing, in the exampleembodiment of FIG. 2 the capacitance between upper back plate 20 andmembrane 22 and between membrane 22 and lower back plate 21, areemployed. A schematic circuit diagram where feedback may be applied withsuch a sensor is schematically shown in FIG. 7. It should be noted thatany quantitative values regarding capacitances, voltages, gains etc.provided in FIG. 7 serve only for illustration purposes and are not tobe construed as limiting, as other embodiments may use other values orother values. Moreover, in other embodiments other circuit structuresmay be used.

In the embodiment of FIG. 7, reference numeral 70 designates a sensor,in the example of FIG. 7 a microphone with two electrodes and a movablemembrane as shown in FIG. 2. INPM and INPP designate connections of thesensor used for reading out the sensor and for variably biasing thesensor, in the embodiment of FIG. 7 for example connections to the twoelectrodes, e.g. back plates.

The connections INPM, INPP are coupled to negative inputs ofdifferential amplifier arrangement 71 and 72, respectively, as shown.Each amplifier arrangement 71, 72 comprises a capacitance and aresistance parallel to the respective amplifier, the capacitance valueof the capacitance being designated C_(F) hereinafter. Furthermore,coupled to INPM, INPP are capacitances 78, the capacitance value ofwhich are designated C_(c) hereinafter and which serve as compensatingcapacitors. Each of the amplifier arrangements 71, 72 forces itscorresponding capacitor plate or electrode, for example plate 20 or 21in FIG. 2, to a voltage supplied to the respective non-inverting inputof the respective operational amplifier, i.e. the input labeled with +in FIG. 7. This input, i.e. first feedback signal is supplied via afirst feedback path comprising an amplifier arrangement 76. Amplifierarrangement 76 may have a gain of one or approximately one. To be ableto sense charge variation due to membrane movement only, i.e. due to theevent to be sensed by the sensor, and to discard an effect coming from aforced differential voltage change, i.e. a voltage change due to thefeedback and the alternately applying of different voltages, a secondfeedback path providing a second feedback signal to the negative inputsof operational amplifier arrangement 71, 72 is provided via thecompensating capacitors 78. An amplifier arrangement 77 drives thecompensating capacitors 78. In an embodiment, the charge compensatingcapacitors 78 and a gain A_(c) of amplifier arrangement 77 are chosensuch thatC ₀ V _(d) −V _(d)(A _(c)−1)C _(c)  (26)

By using a capacitor of the same size as the nominal capacitance ofsensor 70, i.e. the capacitance with no bias voltage applied, forexample a gain of two is selected. To obtain this, for example amplifierarrangement 77 driving the compensating capacitor 78 may have a gain oftwo or approximately two. In some embodiments, C_(c) may be chosen to beslightly greater than C₀ to avoid a transmission zero in the open looptransfer function of the apparatus of FIG. 7.

With the elements of FIG. 7 discussed so far, the displacement of themembrane of sensor 70 is proportional to the sum of the voltages acrossthe capacitances of amplifier arrangement 71, 72. To obtain the sum ofthese voltages which then correspond to the output signal of the sensora differential difference amplifier arrangement 73 is provided. Thedifferential difference amplifier arrangement 73 serves also ascontroller in the feedback loop for the first and second feedback pathsof the embodiment of FIG. 7. In other words, through differentialdifference amplifier arrangement effectively the values for k_(d) andk_(v) mentioned above are determined. An output of differentialdifference amplifier arrangement 73 labeled VMP, VMM is fed to aswitching arrangement 74 which serves for switching between the risingand decaying phase of the arrangement by effectively interchanging theoutputs of differential difference amplifier arrangement 73. Thiseffectively changes the signs of k_(d) and k_(v). In an embodiment, thegain of differential difference amplifier arrangement 73, i.e. k_(d) andk_(v), are selected such that through changing the signs a change ismade between stable and unstable gain settings or, in other words,between rising and decaying phase. The output of switching circuitry 74is supplied via an arrangement 75 to amplifier arrangements 76 and 77,respectively.

The amplifier structure of the embodiment of FIG. 7 may also be referredto as pseudo differential structure.

In the embodiment shown in FIG. 7, the sensor is always operated beyondits so-called pull-in point. In such an embodiment, if for some reasonthe displacement measurement becomes erroneous, for example due to thecapacitances of amplifier arrangement 71, 72 leaking, there is apossibility that the membrane of the sensor will collapse. In someembodiments, to prevent this a refresh cycle is introduced in which thesensor is biased below its pull-in point. This allows the membrane toreturn to its equilibrium position.

While the above embodiment has been described as using displacementfeedback (k_(d)≠0) and velocity feedback (f_(v)≠0), in some embodimentsonly one of these kinds of feedback may be used.

The gain of an embodiment using feedback like the embodiments of FIGS. 6and 7 essentially follows the same general law as explained for the casewithout feedback and as expressed for example in equation (9). However,with feedback the effective time constant τ may be chosen smaller andthus higher gains or higher frequencies may be obtained in someembodiments. However, if the quenching frequency and gain offered by anembodiment without feedback are sufficient for a particular application,an embodiment without feedback may be used as the circuit structure ismore simple. Furthermore, in case with feedback the initial state is not−F_(a)/k′, but

$\begin{matrix}{{x_{0}^{\prime}(0)} = \frac{- F_{a}}{k^{\prime} - {mk}_{d}}} & (27)\end{matrix}$

In other words, the initial displacement caused by the acoustic force incase of a microphone or caused by acceleration or gravity in case of anaccelerometer differs from the case without feedback.

The above-described embodiments are not to be construed as limiting, buthave been provided merely to provide a better understanding of variouspossibilities to implement the present invention. For example, circuitelements shown may be replaced by other elements having the same or asimilar function or, depending on the circumstances, may be omittedaltogether.

What is claimed is:
 1. An apparatus, comprising: a sensor comprising afirst electrode, a movable part, and a second electrode, a firstcapacitance being defined between the first electrode and the movablepart, a second capacitance being defined between the movable part andthe second electrode, the first electrode being directly coupled to anamplifier, and the second electrode being directly coupled to theamplifier; a first switch coupling a first voltage source with: themovable part, or the first electrode and the second electrode; a secondswitch coupling a second voltage source with: the movable part, or thefirst electrode and the second electrode; and a clock, coupled to thefirst switch and the second switch, to alternately close the firstswitch and the second switch.
 2. The apparatus of claim 1, furthercomprising: the amplifier, a first input of the amplifier being coupledto the first electrode, a second input of the amplifier being coupled tothe second electrode, a first output of the amplifier being coupled to afirst output pin, and a second output of the amplifier being coupled toa second output pin.
 3. The apparatus of claim 1, further comprising: atleast one third switch coupled in parallel to the amplifier, the atleast one third switch to operate synchronously with the second switch.4. The apparatus of claim 1, further comprising: at least one thirdcapacitance coupled in parallel with the amplifier.
 5. The apparatus ofclaim 1, further comprising: biasing circuitry coupled with the firstelectrode and the second electrode.
 6. The apparatus of claim 1, whereina voltage of the first voltage source is associated with a stable stateof the movable part, and a voltage of the second voltage source isassociated with an unstable state of the movable part.
 7. The apparatusof claim 1, wherein the first electrode comprises a first plate, thesecond electrode comprises a second plate, and the movable partcomprises a membrane suspended between the first plate and the secondplate.
 8. A method, comprising: receiving, by an apparatus, a firstvoltage from a first voltage source, the apparatus comprising a sensorthat comprises a first electrode, a movable part, and a secondelectrode, a first capacitance being defined between the first electrodeand the movable part, a second capacitance being defined between themovable part and the second electrode, the first electrode beingdirectly coupled to an amplifier, and the second electrode beingdirectly coupled to the amplifier; receiving, by the apparatus, a secondvoltage from a second voltage source; and alternately closing, by theapparatus and based on a clock, a first switch and a second switch, thefirst voltage source being coupled, via the first switch, with: themovable part, or the first electrode and the second electrode, and thesecond voltage source being coupled, via the second switch, with: themovable part, or the first electrode and the second electrode.
 9. Themethod of claim 8, wherein a first input of the amplifier is coupled tothe first electrode, where a second input of the amplifier is coupled tothe second electrode, where a first output of the amplifier is coupledto a first output pin, and where a second output of the amplifier iscoupled to a second output pin.
 10. The method of claim 8, wherein atleast one third switch is coupled in parallel to the amplifier, the atleast one third switch to operate synchronously with the second switch.11. The method of claim 8, wherein at least one third capacitance iscoupled in parallel with the amplifier.
 12. The method of claim 8,wherein biasing circuitry is coupled with the first electrode and thesecond electrode.
 13. The method of claim 8, wherein the first voltageof the first voltage source is associated with a stable state of themovable part, and the second voltage of the second voltage source isassociated with an unstable state of the movable part.
 14. The method ofclaim 8, wherein a membrane, of the movable part, is suspended between afirst plate of the first electrode and a second plate of the secondelectrode.
 15. An apparatus, comprising: a first electrode; a movablepart; and a second electrode, a first capacitance being defined betweenthe first electrode and the movable part, a second capacitance beingdefined between the movable part and the second electrode, the firstelectrode being directly coupled to an amplifier, the second electrodebeing directly coupled to the amplifier, the movable part beingalternately coupled, based on a clock and via a first switch and asecond switch, to a first voltage source and a second voltage source,the movable part being coupled the first voltage source when the firstswitch is closed, and the movable part being coupled to the secondvoltage source when the second switch is closed.
 16. The apparatus ofclaim 15, wherein a first input of the amplifier is coupled to the firstelectrode, and where a second input of the amplifier is coupled to thesecond electrode, where a first output of the amplifier is coupled to afirst output pin, and where a second output of the amplifier is coupledto a second output pin.
 17. The apparatus of claim 15, wherein at leastone third switch is coupled in parallel to the amplifier, the at leastone third switch to operate synchronously with the second switch. 18.The apparatus of claim 15, wherein at least one third capacitance iscoupled in parallel with the amplifier.
 19. The apparatus of claim 15,wherein biasing circuitry is coupled with the first electrode and thesecond electrode.
 20. The apparatus of claim 15, wherein a voltage ofthe first voltage source is associated with a stable state of themovable part, and a voltage of the second voltage source is associatedwith an unstable state of the movable part.